Find an example of a topological space that is seperable and first countable but not second countable. Explain your answer.

Q: If the dual X' of a normed linear space X is fininte dimensional, then X is finite dimensional

A: This is the question of Functional Analysis.

Q: Find an open cover of the interval (−∞, 3]. Justify why that open cover has no finite subcover.

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Q: show that any infinite discrete topological space is not compact

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Q: what is reflexive Banach space ?

A: Banach space: A normed linear space N,· is called banach space if it is complete metric space. A…

Q: Every Hausdorff space is hereditarily Hausdorff.

A: We have to prove that every Hausdorff space is hereditarily Hausdorff. Consider a Hausdorff…

Q: A completely regular space is also T1-space 1 True 2 False

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Q: Let (R,d) be diserete metric space then R is not compact . True or false.??

A: Let (R,d) be diserete metric space then R is not compact . True or false.??

Q: the quotient space of a T0−space necessarily T0

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Q: Define a compact space.

A: The objective is to give the definition of a compact space.

Q: What is norm space

A: In mathematics, a normed space or a normed vector space is a vector space over the real or complex…

Q: . Prove or disprove every metric space is normed space,

A: A normed vector space or normed space is a vector space over the real or complex numbers, on which a…

Q: Define the term dimension of a space.

A: In mathematics, the dimension of space means the minimum number of coordinated to specify or located…

Q: Prove the following: The countable product of metrizable spaces is a metrizable space.

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Q: Show that: All completions of a metric space are isometric.

A: According to the given information, it is required to show that all the completions of a metric…

Q: If X and Y are separable spaces, then X × Y is separable.

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Q: Every T3 space is regular space. True False

A: Since a regular T1 space is called a T3 space.

Q: If X is a Hausdorff topological space, is the quotient space X/∼ necessarily Hausdorff? Explain.

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Q: Please define compact space and the clousure of .a set

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Q: What is the norm of a partition of a closed interval?

A: Concept: The calculus helps in understanding the changes between values that are related by a…

Q: attern on the graph as well as the space provided

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Q: Every inner product space is metric space and normed space O metric space but may be not normed O…

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Q: Let X And Y be two distrecte spaces. Then X is homeomorphic to Y if and only if X and Y are both…

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Q: The quotient topology actually defines a topology.

A: The quotient topology  actually defines a topology

Q: Provide an example of a topological space that is connected but not path-connected,why?

A: We use definition of connected and path connected sets. We use contradiction method.

Q: Let [a, b] be an inter a space. Let ƒ:[a,b] → X

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Q: Is it possible to have a function on a metric space which is discontimuous at every point of the…

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Q: Let X and Y be compact spaces. Then X × Y is compact.

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Q: What is the minimum number of points that can be contained in a space

A: There is no minimum number of points to be contained in a space. But if we want to find a location…

Q: Give an example that satisfies in arbitrary metric space, boundedness and closedness do not…

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Q: What is Euclidean Space??

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Q: Can you show an example of Null space?

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Q: Determine the dimension of the space spanned by the set

A: Given,               S=12-13, 1002, 28-48, 1111, 3306

Q: Iwant to define these Poin (Defined) 1-) sobolve spaces 2-) Reflexive real Banach SPace

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Q: ovide an illustration of the statement "Every normal space need not be a ular space."

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Q: Please define these concepts: normal space, .Hausdorff space

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Q: What is the Boundedness Theorem?

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Q: Define the term Connectedness?

A: In mathematics Connectedness is used to refer to various properties meaning, in some sense, "all one…

Q: a discrete space

A: We know that in a discrete topological space X, every point in X is an open set.

Q: Show that topologically completeness is weak hereditary property.

A: In this question, we have to show that completeness is a weak hereditary property.

Q: topological spaces X and Y,

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Q: Finite topological spaces are always complete. True False

A: Finite topological spaces are always complete. So it is true statement.

Q: (a) Let X and Y be normed spaces and F :X→Y be a continuous linear map. Show that F is closed . Is…

A: Given that,                   X and Y are normed spaces and F : X → Y is a continuous linear map. We…

Q: a) Find a space that is a regular but not Hausdorff. Justify your claim.

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Let (X, τ ) be a discrete space. Is (X, τ ) second countable? Explain why

A: Consider the provided question, Let x,τ be a discrete space,    ⇒x,τ  be denote topology if x,τ is…

Q: Is any discrete space complete? Give an example to substantiate your answer.

A: A metric space is complete if and only if every Cauchy sequence is convergent.

Q: Exhibit an open cover of the interval (1, 2] that has no finite subcover.

A: An open cover of A is a collection of open subsets of A whose union is all of A and that a finite…

Q: Show that the closure of a connected space is connected.

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Q: Let X and Y be two discrete spaces, then *

A: Note: Since you have posted multiple questions we are only allowed to solve the first one. Thankyou…